Question

solve for x 8(x-1/x)^2+2(x+1/x)=121

Answer 1

Hey there!

Given question to us is for finding the values for the variable "x".

Simplify the left hand side.

$\bf{8 \times (\dfrac{x^2 - 2x + 1}{x^2}) + \dfrac{2 \times x + 2 \times 1}{x}}$

$\bf{\dfrac{8x^2 - 16x + 8}{x^2} + \dfrac{2x + 2}{x}}$

Taking the Least common multiplier and simplifying them with a common denominator of "x^2";

$\bf{\dfrac{8x^2 - 16 + 8 + (2x + 2)x}{x^2}}$

$\bf{\dfrac{8x^2 - 16x + 8 + 2x^2 + 2x}{x^2}}$

$\bf{\dfrac{10x^2 - 14x + 8}{x^2}}$

By applying the fractional rule and splitting them into multiple fractions that is,

$\bf{\dfrac{a +- b}{c} = \dfrac{a}{c} +- \dfrac{b}{c}}$

$\bf{\therefore \quad \dfrac{10x^2}{x^2} - \dfrac{14x}{x^2} + \dfrac{8}{x^2}}$

$\bf{\therefore \quad 10 - \dfrac{14}{x} + \dfrac{8}{x^2}}$

Bring back the right hand side for further simplification to find the variable that is,

$\bf{10 - \dfrac{14}{x} + \dfrac{8}{x^2} = 121}$

Take the Least common multiplier of "x^2" again, and multiply this into the numerator;

$\bf{10x^2 - \dfrac{14}{x} x^2 + \dfrac{8}{x^2} x^2 = 121 x^2}$

$\bf{10x^2 - 14x + 8 = 121x^2}$

Subtract by a variable value of "121x^2" on both the sides to further calculate via quadratic formula that is,

$\bf{10x^2 - 14x + 8 - 121x^2 = 121x^2 - 121x^2}$

$\bf{\underline{- 111x^2 - 14x + 8 = 0}}$

For a given quadratic equation of the given form of an equation $\bf{ax^2 + bx + c = 0}$ is providing two sets of given solutions in this type of formula called the quadratic formula that is,

$\boxed{\bf{x_{1, \: 2} = \dfrac{- b +- \sqrt{b^2 - 4ac}}{2a}}}$

Here, a = - 111,  b = - 14  and c = 8.

$\bf{\therefore \quad x_1 = \dfrac{- (- 14) + \sqrt{(- 14)^2 - 4(- 111)(8)}}{2 (- 111)}}$

$\bf{\therefore \quad x_1 = \dfrac{14 + \sqrt{196 + 3552}}{- 2 \times 111}}$

$\bf{\therefore \quad x_1 = \dfrac{14 + \sqrt{3748}}{- 2 \times 111}}$

$\bf{\therefore \quad x_1 = \dfrac{14 + \sqrt{3748}}{- 222}}$

$\bf{\therefore \quad x_1 = \dfrac{14 + \sqrt{2 \times 2 \times 937}}{222}}$

$\bf{\therefore \quad x_1 = - \dfrac{14 + \sqrt{2^2 \times 937}}{222}}$

$\bf{\therefore \quad x_1 = - \dfrac{14 + 2 \sqrt{937}}{222}}$

$\boxed{\bf{\therefore \quad x_1 = - \dfrac{7 + \sqrt{937}}{111}}}$

AND, FOR THE NEGATIVE VALUE OR THE SECOND VALUE FOR "x" THAT IS "X_2":

$\bf{\therefore \quad x_2 = \dfrac{- (- 14) - \sqrt{(- 14)^2 - 4(- 111)(8)}}{2 (- 111)}}$

$\bf{\therefore \quad x_2 = \dfrac{14 - \sqrt{196 + 3552}}{- 2 \times 111}}$

$\bf{\therefore \quad x_2 = \dfrac{14 - \sqrt{3748}}{- 2 \times 111}}$

$\bf{\therefore \quad x_2 = \dfrac{14 - \sqrt{3748}}{- 222}}$

$\bf{\therefore \quad x_2 = \dfrac{\sqrt{2 \times 2 \times 937} - 14}{222}}$

$\bf{\therefore \quad x_2 = \dfrac{\sqrt{2^2 \times 937} - 14}{222}}$

$\bf{\therefore \quad x_2 = \dfrac{2 \sqrt{937} - 14}{222}}$

$\boxed{\bf{\therefore \quad x_2 = \dfrac{\sqrt{937} - 7}{111}}}$

Therefore, the required solutions obtained for the variable "x" for this equation are:

$\boxed{\bf{\underline{\therefore \quad x_1 = - \dfrac{7 + \sqrt{937}}{111}}}}$

$\boxed{\bf{\underline{\therefore \quad x_2 = \dfrac{\sqrt{937} - 7}{111}}}}$

Which are the required solutions for these types of queries.

Hope this helps you and clears your doubt for finding the values of given variables via quadratic formula after simplification of equations !!!!!!!!